Naturalness, Weak Scale Supersymmetry and the Prospect

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The above general connection would be thwarted if there were large internal cancellations occurring naturally within the radiative breaking condition which would allow m0 and m1/2 disproportionately large for a fixed fine tuning. We shall show that precisely such a situation does arise in certain domains of the supergravity parameter space. The simplest fine tuning criterion is to impose the constraint that m0,mg˜ < 1 TeV where m0 is the universal soft SUSY breaking scalar mass in minimal supergravity and mg˜ is the gluino mass. The above criterion is easy to implement and has been used widely in the literature (for a review see Ref.[3]). A more involved fine tuning criterion is given in Ref.[4]. However, it appears that the criterion of Ref.[4] is actually a measure of the sensitivity rather than of fine tuning[5, 6]. Another naturalness criterion is proposed in Ref.[6] and involves a distribution function. Although the distribution function is arbitrary the authors show that different choices of the function lead numerically to similar fine tuning limits. In the analysis of this paper we use the fine tuning criterion introduced in Ref.[7] in terms of the Higgs mixing parameter µ which has several attractive features. It is physically well motivated, free of ambiguities and easy to implement. Next we use the criterion to analyze the upper limits of sparticle masses for low values of tanβ, i.e., tanβ 5. In this case one finds that m0 and m1 2 allowed by radiative breaking lie ≤ / on the surface of an ellipsoid, and hence the upper limits of the sparticle masses are directly controlled by the radii of the ellipsoid which in turn are determined by the choice of fine tuning. For instance, one finds that if one is in the low tanβ end of b τ − unification[8] with the top mass in the experimental range, i.e. tanβ 2, then for ≈ any reasonable range of fine tuning the sparticle mass upper limits for the entire set of SUSY particles lie within the mass range below 1 TeV. Further, one finds that the light Higgs mass lies below 90 GeV under the same constraints. Thus in this case discovery of supersymmetry at the LHC is guaranteed according to any reasonable fine tuning criterion. Next the paper explores larger values of tanβ, i.e., tanβ 10 and here one ≥ finds that m0 and m1/2 for moderate values of fine tuning do not lie on the surface of an ellipsoid; rather one finds that they lie on the surface of a hyperboloid. In this case m0 and m1/2 are not bounded by the µ constraint equation and large values of m0 and m1/2 can result with a fixed fine tuning. Effect of non-universalities on naturalness is also analyzed. Again one finds phe- nomena similar to the ones discussed above, although the domains in which these phe- nomena occur are shifted relative to those in the universal case. One of the important results that emerges is that the upper limits of sparticle masses can be dramatically affected by non-universalities. These results have important implications for the discovery of supersymmetry at the Tevatron and the LHC. Our analysis is carried out in the framework of supergravity models with gravity mediated breaking of supersymmetry[9, 1, 3]. This class of models possesses many attractive features. One of the more attractive features of these models is that with R parity invariance the lightest neutralino is also the lightest supersymmetric particle over most of the parameter space of the theory and hence a candidate for cold dark matter. Precision renormalization group analyses show[10] that these models can accommodate

2 just the right amount of dark matter consistent with the current astrophysical data[11, 12]. However, in this work we shall not impose the constraint of dark matter. The outline of the paper is as follows: In Sec.2 we give a brief discussion of the fine tuning measure used in the analysis. In Sec.3 we use this criterion to discuss the upper limits on the sparticle masses in minimal supergravity for low tanβ, i.e., tanβ 5 and show that the allowed solutions to radiative breaking lie on the surface ≤ of an ellipsoid. In Sec.4 we discuss naturalness in beyond the low tanβ region. Here we show that radiative breaking of the electro-weak symmetry leads to the soft SUSY breaking parameters lying on the surface of a hyperboloid. In Sec.5 we discuss the effects of non-universalities on the upper limits. In Sec.6 we show that a high degree of fine tuning is needed to have the light Higgs mass approach its maximum upper limit. The limits on Φ from the current data are discussed in sec.7. Implications of these results for the discovery of supersymmetric particles at colliders is also discussed in Secs. 3-6. Conclusions are given in Sec.8.

2 Measure of Naturalness

We give below an improved version of the analysis of the ﬁne tuning criterion given in Ref.[7]. The radiative electro-weak symmetry breaking condition is given by

1 2 2 2 M = λ µ (1) 2 Z − where λ2 is defined by 2 2 2 2 m¯ m¯ tan β λ = H1 − H2 (2) tan2 β 1 − 2 2 Herem ¯ Hi = mHi + Σi(i=1,2) where Σi arise from the one loop corrections to the effective potential[13]. The issue of fine tuning now revolves around the fact that a cancellation is needed between the λ2 term and the µ2 term to arrange the correct 2 experimental value of MZ . Thus a large value of λ would require a large cancellation from the µ2 term resulting in a large fine tuning. This idea can be quantified by defining the fine tuning parameter Φ so that

2 2 −1 λ µ Φ = 4 − (3) λ2 + µ2

(The factor of 4 on the right hand side in Eq.(3) is just a convenient normalization.) The expression for Φ can be simplified by inserting in the radiative breaking condition Eq.(1). We then get 1 µ2 Φ= + 2 (4) 4 MZ The result above is valid with the inclusion of both the tree and the loop corrections to the effective potential.(Φ is related to the fine tuning parameter δ defined in Ref.[7] −1 µ2 by Φ = δ ). For large µ one has Φ M 2 , a result which has a very direct intuitive ∼ Z meaning. A large µ implies a large cancellation between the λ2 term and the µ2 term

3 in Eq.1 to recover the Z boson mass and thus leads to a large ﬁne tuning. Typically a large µ implies large values for the soft supersymmetry breaking parameters m0 and m1/2 and thus large values for the sparticle masses. However, large cancellation can be enforced by the internal dynamics of radiative breaking itself. In this case a small µ and hence a small ﬁne tuning allows for relatively large values of m0 and of m1/2. We show that precisely such a situation arises for certain regions of the parameter space of both the minimal model as well as for models with non-universalities.

3 Upper Bounds on Sparticle Masses in Mini- mal Supergravity

We discuss now the upper bounds on the sparticle masses that arise under the criterion of fine tuning we have discussed above. Using the radiative electro-weak symmetry breaking constraint and ignoring the b-quark couplings, justified for small tanβ, we may express the fine tuning parameter Φ0 in the form

2 m 1 m 1 A0 1 m0 2 A0 2 2 2 2 ∆µloop Φ0 = + ( ) C1 + ( ) C2 + ( ) C3 + ( 2 )C4 + 2 (5) −4 MZ MZ MZ MZ MZ where 2 1 3D0 1 2 t C1 = (1 − t ),C2 = k (6) t2 1 − 2 t2 1 − − 2 2 1 2 t 2 Σ1 t Σ2 C3 = (g t e), C4 = f, ∆µ = − (7) t2 1 − −t2 1 loop t2 1 − − − Here t tanβ, e,f,g,k and the sign conventions of A0 and µ are as defined in Ref.[14], ≡ D0 is defined by 2 D0 = 1 (m /m ) , m 200sinβ GeV (8) − t f f ≃ and Σ1 and Σ2 are as defined in Ref. [13]. To investigate the upper limits on m0 and m1/2 consistent with a given fine tuning it is instructive to write Eq.(5) in the form

2 ′2 ′ 2 2 2 1 C1m0 + C3m1 2 + C2A0 +∆µ = M (Φ0 + ) (9) / loop Z 4 where 2 ′ 1 C4 ′ 1 C4 m1/2 = m1/2 + A0 , C2 = C2 (10) 2 C3 − 4 C3 2 and ∆µloop is the loop correction. Now for the universal case one finds that the loop corrections to µ are generally small for tanβ 5 in the region of fine tuning of Φ0 20. ≤ ′ ≤ Further, using renormalization group analysis one finds that C2 > 0 and C3 > 0 and at least for the range of fine tuning Φ0 20, C1 > 0(see Table 1). Thus in this case ≤ defining 1 1 1 2 2 Φ+ 4 2 2 Φ+ 4 2 2 Φ+ 4 a = MZ , b = MZ , c = MZ ′ (11) C3 C1 C2

4 we find that for tanβ 5, Φ0 20 the radiative breaking condition can be approxi- ≤ ≤ mated by ′2 2 2 m1 2 m0 A0 / + + 1 (12) a2 b2 c2 ≃ and the renormalization group analysis shows that at the scale Q = MZ the quantities 2 2 2 a , b an c are positive. Fixing the fine tuning parameter Φ0 fixes a,b, and c and one finds that m0 and m1/2 are bounded as they lie on the boundary of an ellipse. Further Eq.(12) implies that the upper bounds on m0 and m1 2 increase as √Φ0 for large / ∼ Φ0. A similar dependence on fine tuning was observed in the analysis of ref. 4. We give now the full analysis without the approximation of Eq.(12). We consider the case of tanβ = 2 first which lies close to the low end of the tanβ region of b-τ unification with the top mass taken to lie in the experimental range[8]. In Fig.1a. we give the contour plot of the upper limits for the parameters m0 and m1 2 in the m0 m1 2 / − / plane for the case of tanβ = 2 and m = 175 GeV for 2.5 Φ0 20. As expected, one t ≤ ≤ finds that the contours corresponding to larger values of m0 and m1/2 require larger values of Φ0. The upper limits of the mass spectra for the same set of parameters as in Fig.1a are analyzed in Fig.1b - Fig.1d. In Fig.1b the upper limits of the mass spectra of the heavy Higgs, the first two generation squarks, and the gluino are given. We find that the mass of the squark and of the gluino are very similar over essentially the entire range of Φ0. Upper limits ofe ˜, t˜1, t˜2 are given in Fig.1c. In Fig.1d we exhibit the upper limits for the light Higgs, the chargino, and the lightest neutralino. We note that 1 except for small values of Φ0 one finds that the scaling laws [15](e.g. m 0 2 m ± ) χ1 ≃ χ1 are obeyed with a high degree of accuracy. We note that the Higgs mass upper limit in this case falls below 85-90 GeV for Φ0 20. At the Tevatron in the Main Injector era ≤ one will be able to detect charginos using the trileptonic signal[16] with masses up to 230 GeV with 10fb−1 of integrated luminosity[17, 18]. Reference to Fig.1d shows that the above implies that the upper limit of chargino masses for the full range of Φ0 20 ≤ will be accessible at the Tevatron. For the gluino the mass range up to 450 GeV will be accessible at the Tevatron in the Main Injector era with 25fb−1 of integrated luminosity. This means that one can explore gluino mass limits up to Φ0=10 for tanβ = 2. However, at the LHC gluino masses in the range 1.6-2.3 TeV [19]/1.4-2.6 TeV[20] for most values of µ and tanβ will be accessible and recent analyses show that the accuracy of the mg˜ mass measurement can be quite good, i.e., to within 1-10% depending on what part of the supergravity parameter space one is in[21]. Thus for tanβ = 2 one will be able to observe and measure with reasonable accuracy the masses of the charginos, the gluino, and the squarks for the full range of values of Φ0 20 at the LHC. It has recently been argued ≤ that the NLC, where even more accurate mass measurements[22, 23, 24] are possible, will allow one to use this device for the exploration of physics at the post-GUT and string scales[25]. The NLC also offers the possibility of testing a good part of the parameter space for the tanβ = 2 model. The analysis given in Table 2 shows that the full sparticle mass spectrum for tanβ=2 can be tested at the NLC with √s = 1 TeV for Φ0 10 and over the entire range Φ0 20 with √s = 1.5 TeV. ≤ ≤ We discuss next the upper limit of sparticle masses for tanβ = 5. In Fig.2a we give the contour plot of m0 and m1 2 upper limits in the m0 m1 2 plane for the same value / − / 5 of the top mass and in the same Φ0 range as in Fig.1a. Here we find that for fixed Φ0 the contours are significantly further outwards compared to the case for tanβ = 2. Correspondingly the upper limits of the mass spectra for the same value of Φ0 are significantly larger in Fig. 2b-2d relative to those given in Figs. 1b-1d. In this case the light chargino mass lies below 243 GeV for Φ0 20 and thus the upper limits for values ≤ of Φ0 20 could be probed at the Tevatron in the Main Injector era where chargino ≤ −1 masses up to 280 GeV will be accessible with 100fb of integrated luminosity[17, 18]. Similarly in this case the gluino mass lies below 873 GeV for Φ0 20 and thus the ≤ upper limit for values of Φ0 20 could be probed at the LHC which as mentioned above ≤ can probe gluino masses in the mass range of 1.6 2.3 TeV[19]/ 1.4 2.6 TeV[20]. LHC − − can probe squark masses up to 2-2.5 TeV, so squark masses of the above size should be accessible at the LHC. A full summary of the results for values of tanβ=2-20 is given in Table 2 where the sparticle mass limits in the range 2.5 Φ0 20 are given. The ≤ ≤ analysis tells us that for a reasonable constraint on Φ0, i.e. Φ0 20, the gluino and ≤ the squarks must be discovered at the LHC for the values of tanβ 5. ≤ 4 Regions Of the Hyperbolic Constraint

In this section we discuss the possibility that in certain regions of the supergravity parameter space the sparticle spectrum can get large even for modest values of the fine tuning parameter Φ0. This generally happens in regions where the loop corrections to µ are large. For example, in contrast to the case of small tanβ one finds that for the case of large tanβ the loop corrections to µ can become rather significant. In this case the size of the loop corrections to µ depends sharply on the scale Q0 where the minimization of the effective potential is carried out. In fact, in this case there is generally a strong dependence on Q0 of both the tree and the loop contributions to µ which, however, largely cancel in the sum, leaving the total µ with a sharply reduced but still non-negligible residual Q0 dependence. An illustration of this phenomenon is given in Fig.3. The choice of Q0 where one carries out the minimization of the effective potential is of importance because we can choose a value of Q0 where the loop corrections are small so that we can carry out an analytic analysis similar to the one in Sec.3. (For example, for the case of Fig. 3 the loop correction to µ is minimized at Q0 1 TeV). Generally we find the value Q0 at which the loop correction to µ is ≈ minimized to be about the average of the smallest and the largest sparticle masses, a ˜ ˜ value not too distant from √mtL mtR , which is typically chosen to minimize the 2-loop correction to the Higgs mass[26, 12]. Choosing a value Q0 where the loop correction is small (Q0 is typically greater than 1 TeV here), and following the same procedure as in Sec.3 we find that this time sign(C1(Q0))=-1 (see entries for the case tanβ = 10, 20 in Table 1 and see also fig. 5a). There are now two distinct possibilities: case A and case B which we discuss below.

Case A: This case corresponds to

1 2 ′ 2 (Φ0 + )M C2A0 > 0 (13) 4 Z −

6 and occurs for relatively small values of A0 . Here the radiative breaking equation | | takes the form ′2 2 m1/2 m0 2 2 1 (14) α (Q0) − β (Q0) ≃ where 1 2 ′ 2 2 (Φ0 + )M C2A0 α = | 4 Z − | (15) C3 | | and 1 2 ′ 2 2 (Φ0 + )M C2A0 β = | 4 Z − | (16) C1 | | The appearance of a minus sign changes intrinsically the character of the constraint of the electro-weak symmetry breaking. One finds now that unlike the previous case, where m0 and m1/2 lie on the boundary of an ellipse for fixed A0 (see Fig.4a and also Figs.1a and 2a), here they lie on a hyperbola. A diagrammatic representation of the constraint of Eq.(14) is given in Fig.4b-4c. The position of the apex of the hyperbola depends on A0 as can be seen from Fig.4c. The choice of Φ itself does not put an upper bound on m0 and m1/2 and consequently they can get large for a fixed fine tuning unless other constraints intervene. Thus in this case the rule that the upper bounds are proportional to √Φ0 breaks down. In fact from Eq.(14)-(16) we see that for large m0 and m1/2 one has

C3 ′ m0 | |m1/2 (17) ≃ s C1 | | and thus independent of Φ0. Thus the hyperbolae for diﬀerent values of ﬁne tuning have the same asymptote independent of Φ0 as illustrated in Fig.4b.

Case B: This case corresponds to

1 2 ′ 2 (Φ0 + )M C2A0 < 0 (18) 4 Z − and occurs for relatively large values of A0. Here the radiative breaking equation takes the form 2 ′2 m0 m1/2 2 2 1 (19) β (Q0) − α (Q0) ≃ A diagrammatic representation of this case is given in Fig 4d. As in Case A, here also m0 and m1/2 lie on a hyperbola, with the position of the apex determined by the value of A0. Again here as in case A the choice of Φ0 itself does not control the upper bound on m0 and m1/2. This can be seen from Fig.4d where the hyperbolae for different values of the fine tuning have the same asymptote independent of Φ0 just as in case A. We emphasize that the analytic analysis based on Eqs.(14) and (19) is for illustrative purposes only, and the results presented in this paper are obtained including the b- quark couplings and including the full one loop corrections to µ. In Fig.5b we present a numerical analysis of the allowed region of m0 and m1/2. One finds that the cases A0 = 0 and A0 = 500 GeV show that m0 and m1/2 lie on a branch of a hyperbola

7 and simulate the illustration of Fig.4c. This is what one expects for the small A0 case. Similarly for the cases A0 = 1000 GeV and A0 = 2000 GeV in Fig.5b, m0 and − − m1/2 lie on a branch of a hyperbola and simulate the illustration of the right hyperbola in Fig.4d as is appropriate for a large negative A0. Similarly for the case A0 = 1000 GeV in Fig.5b, m0 and m1/2 again lie on a branch of a hyperbola and simulate the illustration of the left hyperbola in Fig.4d. A similar analysis for tanβ = 20 can be found in Fig.5c. Thus one ﬁnds that the results of the analytic analysis are supported by the full numerical analysis.

5 Eﬀects of Non-universal Soft SUSY Breaking

The analysis of Secs. 3 and 4 above is carried out under the assumption of universal soft supersymmetry boundary conditions at the GUT scale. These universal boundary conditions arise from the assumption of a flat Kahler potential. However, the framework of supergravity unification[1, 3] allows for more general Kahler structures and hence for non-universalities in the soft supersymmetry breaking parameters[27, 25]. In the analysis of this section we shall assume universalities in the soft supersymmetry breaking parameters in the first two generations of matter but allow for non-universalities in the Higgs sector[25, 28, 29, 30] and in the third generation of matter[25, 30, 31]. It is convenient to parametrize the non-universalities in the following fashion. In the Higgs sector one has

2 2 2 2 mH1 = m0(1 + δ1), mH2 = m0(1 + δ2) (20) Similarly in the third generation sector one has

2 2 2 2 m ˜ = m0(1 + δ3), m ˜ = m0(1 + δ4) (21) QL UR A reasonable range for the non-universality parameters is δ 1 (i=1-4). Inclusion of | i|≤ non-universalities modifies the electro-weak symmetry breaking equation determining the parameter µ2, and leads to corrections to the fine tuning parameter Φ. One finds that with these non-universality corrections Φ is given by

2 m 1 m 1 A0 1 m0 2 ′ A0 2 2 2 2 ∆µloop Φ= + ( ) C1 + ( ) C2 + ( ) C3 + ( 2 )C4 + 2 (22) −4 MZ MZ MZ MZ MZ where

′ 1 3D0 1 2 1 2 D0 1 2 C1 = (1 − t )+ (δ1 δ2t − (δ2 + δ3 + δ4)t ) t2 1 − 2 t2 1 − − 2 − − 2 3 t + 1 pS0 + 2 2 (23) 5 t 1 m0 − and C2, C3 and C4 are as deﬁned in Eqs.(6) and (7). Here S0 is the trace anomaly term

2 S0 = T r(Ym ) (24)

8 evaluated at the GUT scale MG. It vanishes in the universal case since Tr(Y)=0, but contributes when non-universalities are present. p is as deﬁned in Ref. [30]. Numerically for M = 1016.2 GeV and α = 1/24 one has p 0.045. Eq.(23) shows G G ≃ how important the eﬀects of non-universalities are on Φ. For a moderate value of 2 m0 = 250 GeV the factor (m0/M ) is 7.5 and since δ O(1), Φ gets a huge shift. Z ∼ i ∼ This means that the upper limits of the sparticle masses are going to be sensitively dependent on the magnitudes and signatures of δi. It is instructive to write the radiative breaking equation Eq.(21) with non-universalities in a form similar to Eq.(9). We get

′ 2 ′2 ′ 2 2 2 1 C1m0 + C3m1 2 + C2A0 +∆µ = M (Φ + ) (25) / loop Z 4 ′ 2 where C1 is defined in Eq.(23), and C2 and C3 are defined in Eq.(7) and where ∆µloop is the loop correction. We discuss the case of non-universalities in the Higgs sector first and consider two extreme examples within the constraint of δ 1(i=1-2). These are | i|≤ (i) δ1 = 1, δ2 = 1, and (ii)δ1 = 1, δ2 = 1, with δ3 =0= δ4 in both cases. For − − case(i) we find from Eq.(23) that the non-universalities make a positive contribution ′ ′ to C1, and thus C1 > 0 (see Table 3). As for the universal case the loop corrections in this case are generally small. Thus in this case one finds that the radiative breaking condition takes the form ′2 2 2 m1 2 m0 A0 / + + 1 (26) a2 b′2 c2 ≃ where a and c are defined by Eq.(11) and b′ is defined by 1 ′2 2 (Φ + 4 ) b = MZ ′ , (27) C1 | | As in the universal case (see Figs.1a, 2a and 4a) here also for given fine tuning one finds that m0 and m1/2 are bounded as they lie on the boundary of an ellipse. Further, ′ C1 > C1 implies that a given Φ corresponds effectively to a smaller Φ0, and hence admits smaller values of the upper limits of the squark masses relative to the universal case. This is what is seen in Table 4. Here we find that the upper limits are generally decreased over the full range of Φ. For case(ii) the situation is drastically different. Here the non-universalities make ′ ′ a negative contribution driving C1 negative (see Table 3) and further C1 remains negative in the relevant Q range (see Table 5). Thus the radiative breaking solutions no longer lie on the boundary of an ellipse. The analysis in this case is somewhat more 2 complicated in that the loop corrections to µ at the scale Q = MZ are large. For illustrative purposes one may carry out an analysis similar to the one discussed in Sec.3 ′ 2 and go to the scale Q=Q0, where the loop corrections to µ are negligible. Again there are two cases and we discuss these below.

Case C: This case is deﬁned by Eq.(13) and the radiative symmetry breaking constraint here reads ′2 2 m1/2 m0 2 ′ ′2 ′ 1 (28) α (Q0) − β (Q0) ≃

9 where 1 2 ′ 2 ′2 (Φ + 4 )MZ C2A0 β = | ′ − | (29) C1 | | Eq.(28) shows that the radiative symmetry breaking constraint in this case is a hyperbolic constraint. Case D: This case is defined by Eq.(18) and the radiative symmetry breaking constraint here reads 2 ′2 m0 m1/2 ′2 ′ 2 ′ 1 (30) β (Q0) − α (Q0) ≃ Again the radiative symmetry breaking constraint is a hyperbolic constraint. Cases C and D are similar to the cases A and B except that here m0 and m1/2 lie on a hyperbola even for small tanβ because of the effect of the specific nature of the non-universalities in this case. Thus here it is the non-universalities which transform the radiative breaking equation from an ellipse to a hyperbola. Of course m0 and m1/2 do not become arbitrarily large, since eventually other constraints set in and limit the allowed values of m0 and m1/2. Results of the analysis are given in Fig.(6). One finds that m0 and m1/2 indeed can become large for a fixed fine tuning. To understand the effects of the non-universalities in the third generation in comparison to the non-universalities in the Higgs sector it is useful to express ∆Φ in the following alternate form

2 1 1 mt 2 2 1 mt 2 2 m0 2 3 t + 1 pS0 ∆Φ = (δ1 (1 ( ) )δ2t + ( ) (δ3 + δ4)t )( ) + (31) t2 1 − − 2 m 2 m M 5 t2 1 M 2 − f f Z − Z

1 mt 2 Since mt < mf one has (1 2 ( ) ) > 0 which implies that the effect of a nega- − mf tive(positive) δ2 can be simulated by a positive(negative) value of δ3 or by a positive(negative) value of δ4. This correlation can be seen to hold by a comparison of Tables 4 and 6. As in the case of Table 4 where a positive δ1 and a negative δ2 leads to lowering of the upper limits on squark masses, we find that a positive δ3 or a positive δ4 produces a similar effect. The analysis of Table 6 where we choose (δ1, δ2, δ3, δ4)=(0,0,1,0) supports this observation. A similar correlation can be made between the case of δ1 < 0, δ2 > 0 and the case δ3 + δ4 < 0 by the comparison given above. We note, however, that the effects of non-universalities in the Higgs sector and in the third generation sector are not identical in every respect as they enter in different ways in other parts of the spectrum. However, the gross features of the upper limits of squarks in Table 6 can be understood by the rough comparison given above. A comparison of Tables 2,4, and 6 shows that the non-universalities have a remark- able effect on the upper limits of sparticle masses. One finds that the upper limits on the sparticle masses can increase or decrease dramatically depending on the type of non-universality included in the analysis. The prospects for the observation of sparticles at colliders are thus significantly affected. For the case of Table 4 and 6 one finds that the sparticle spectrum falls below 1 TeV in the range tanβ 5, Φ 20. Thus ≤ ≤ in this case the gluino and the squarks should be discovered at the LHC and all of the other sparticles should also be discovered over most of the mass ranges in Table 4.

10 In contrast for the case of non-universality of Table.6 we ﬁnd that the nature of non- universal contribution is such that squark masses can exceed the discovery potential of even the LHC. The analysis given above is for µ< 0. A similar analysis holds with essentially the same general conclusions for the µ> 0 case.

6 Upper Limit on the Higgs Mass

One of the most interesting part of our analysis concerns the dependence of the Higgs mass upper limits on Φ0. For the analysis of the Higgs mass upper limits we have taken account of the one loop corrections to the masses and further chosen the scale Q which minimizes the two loop corrections[26, 12]. For tanβ = 2 the upper limit on the Higgs mass increases from 60 GeV at Φ0=2.5 GeV to 86 GeV at Φ0=20. Further from the successive entries in this case in Table 2 we observe that in each of the cases where an increment in the Higgs mass occurs, one requires a significant increase in the value of Φ0. The same general pattern is repeated for larger values of tanβ. Thus for tanβ = 5 the Higgs mass increases from 97 GeV to 116 GeV as Φ0 increases from 2.5 to 20. In Fig.7 we exhibit the upper bound on the Higgs mass as a function of tanβ. From the analysis of Table 2 and Fig.7 one can draw the general conclusion that the Higgs mass upper limit is a sensitive function of tanβ and Φ0. For values of tanβ near the low end, i.e. tanβ 2, the upper limit of the Higgs mass lies below 85-90 GeV for ≈ any reasonable range of fine tuning, i.e. Φ0 20. This is a rather strong result. Thus ≤ if the low tanβ region of b τ unification turns out to be the correct scenario then − our analysis implies the existence of a Higgs mass below 85-90 GeV for any reasonable range of fine tuning. This scenario will be completely tested at LEPII which can allow coverage of the Higgs mass up to m 95 GeV with √s = 192 GeV. If no Higgs is h ≈ seen at LEPII then a high degree of fine tuning, i.e. Φ0 > 20, is indicated on the low tanβ end of b τ unification. − Further, the analysis also indicates that in order to approach the maximum allowed Higgs mass one needs to have a high degree of fine tuning. In particular from Table 2 and Fig.7 we see that going beyond 120 GeV in the Higgs mass requires a value of Φ0 on the high side, preferably 10 and 20. The strong correlation of the Higgs mass upper limits with the value of Φ0 has important implications for sparticle masses. Thus if the Higgs mass turns out to lie close to its allowed upper limit then a larger value of Φ0 would be indicated. In turn a large Φ0 would point to a heavy sparticle spectrum. At TeV33 with 25fb−1 of integrated luminosity Higgs mass up to 120 GeV will be probed. A non-observation of the light Higgs in this mass range will imply that one needs a high degree of fine tuning which would point in the direction of heavy sparticle masses. These results are in general agreement with the analysis of Ref.[32] which arrived at much the same conclusion using a very different criterion of fine tuning. In particular the analysis of Ref.[32] also found that the non-observation of the Higgs mass below 120 GeV will imply a heavy spectrum.

11 7 Fine-tuning limits from the current experimental data

One may put limits on the fine tuning parameter using the current experimental data on sparticle searches at colliders[33, 34]. The result of this analysis is presented in Table 7. For low tanβ the strongest lower limits on the fine tuning parameter arise from the lower limits on the Higgs mass. In Table 7 we have used the experimental lower limits on the Higgs mass from the four detectors at LEP, i.e., the L3, OPAL, ALEPH, and DELPHI[33], to obtain lower limits on Φ for values of tanβ from 2 to 20. As expected one finds that the strongest limit on Φ arises for the smallest tanβ, and the constraint on Φ falls rapidly for larger tanβ. Thus for tanβ greater than 5 the lower limit on Φ already drops below 2 which is not a stringent fine tuning constraint. Lower limits on Φ from the current data on the lower limits on the neutralino, the chargino, the stop, the heavy squarks, and the gluino are also analysed in Table 7. One finds that here the current lower limits on the chargino mass produce the stongest lower limit on Φ. For tanβ of 2, the lower limit on Φ from the Higgs sector is still more stringent constraint than the lower limit constraint from the chargino sector. However, for tanβ=5 the constraint from the chargino sector becomes more stringent than the constraint from the Higgs sector. These constraints on the fine tuning will become even more stringent after LEP II completes its runs and if supersymmetric particles do not become visible.

8 Conclusions

In this paper we have analyzed the naturalness bounds on sparticle masses within the framework of radiative breaking of the electro-weak symmetry for minimal supergravity models and for non-minimal models with non-universal soft SUSY breaking terms. For the case of minimal supergravity it is found that for small values of tanβ, i.e., tanβ 5 ≤ and a reasonable range of fine tuning, i.e., Φ 20, the allowed values of m0 and m1 2 ≤ / lie on the surface of an ellipsoid with the radii determined by the value of fine tuning. Specifically for the case tanβ = 2 it is found that the upper limits on the gluino and squark masses in minimal supergravity lie within 1 TeV and the light Higgs mass lies below 90 GeV for Φ0 20. For tanβ 5 the upper limits of the sparticle masses all ≤ ≤ still lie within the reach of the LHC for the same range of Φ0. The analysis shows that the upper limits of sparticle masses are very sensitive functions of tanβ. As values of tanβ become large the loop corrections to µ become large and the nature of the radiative breaking equation can change, i.e., m0 and m1/2 may not lie on the surface of an ellipsoid. Thus it is found that there exist regions of the parameter space for large tanβ where the upper bounds on the sparticle masses can get very large even for reasonable values of fine tuning. We have also analyzed the effects of non-universalities in the Higgs sector and in the third generation sector on the upper limits on the sparticle masses. It is found that non-universalities have a very significant effect on the overall size of the sparticle mass upper limits. Thus we find that the case (i) δ1 > 0 or δ2 < 0 and δ3 =0= δ4 has the effect of decreasing the upper limits on the squark masses, and in contrast

12 the case (ii) δ1 < 0 or δ2 > 0 and δ3 =0= δ4 has the effect of increasing the upper limits on the squark masses. Remarkably for δ1 = 1, δ2 = 1 and δ3 =0= δ4 all − of the sparticle masses lie below 1 TeV for tanβ 5 and Φ 20 because of the non- ≤ ≤ universality effects. In this case the sparticles would not escape detection at the LHC. However, for the case δ1 = 1, δ2 = 1 and δ3 =0= δ4 there is an opposite effect and − the non-universalities raise the upper limits of the sparticle masses. Here for the same range of tanβ, i.e., tanβ 5 the first and second generation squark masses can reach ≤ approximately 3 TeV for Φ 10 (4-5 TeV for Φ 20) and consequently these sparticles ≤ ≤ may escape detection even at the LHC. Similar effects occur for the non-universalities in the third generation sector. Thus non-universalities have important implications for the detection of supersymmetry at colliders. Finally, it is found that the upper limit on the Higgs mass is a very sensitive function of tanβ in the region of low tanβ and moving the upper limit beyond 120 GeV towards its maximally allowed value will require a high degree of fine tuning. In turn large fine tuning would result in a corresponding upward movement of the upper limits of other sparticle masses. Thus a non-observation of the Higgs at the upgraded Tevatron with an integrated luminosity of 25fb−1, would imply a high degree of fine tuning and point to the possibility of a heavy sparticle spectrum. Acknowledgements Fruitful discussions with Richard Arnowitt, Howard Baer and Haim Goldberg are ac- knowledged. This research was supported in part by NSF grant number PHY-96020274.

13 References

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15 [31] Effects of non-universalities on the upper limits of sparticle masses in the context of the fine tuning criterion of Ref.[4] is discussed in S. Dimopoulos and G. Giudice, Phys. Lett.B357, 573(1995). [32] G.W. Anderson, D.J.Castaño and A. Riotto, Phys. Rev. D55, 2950(1997). [33] Joachim Mnich, LEP Physics - An Overview, plenary talk at The Sixth Interna- tional Symposium on PARTICLES, STRINGS and COSMOLOGY, PASCOS-98, Boston, Massachusetts, March 22-29, 1998; Joachim Mnich, Results of the L3 Experiment at 183 GeV, presentation at CERN LEPC, 31 March 1998; John Carr, ALEPH Status Report, presentation at CERN LEPC, 31 March 1998; Klaus Moenig, DELPHI Results at 183 GeV, presentation at CERN LEPC, 31 March 1998; Mark Thomson, OPAL Physics Highlights at √s= 183 GeV, presentation at CERN LEPC, 31 March 1998. [34] http://wwwd0.fnal.gov/public/new public.html.

16 Scale dependence of C1 – C4 tan β Q(GeV ) C1 C2 C3 C4 2 91.2 0.7571 0.0711 4.284 0.3119 2000 0.6874 0.0879 2.851 0.3073 4000 0.6702 0.0918 2.607 0.3055 6000 0.6598 0.0941 2.474 0.3043 8000 0.6523 0.0957 2.384 0.3034 10000 0.6464 0.0970 2.316 0.3026 5 91.2 0.14212 0.1024 2.871 0.4491 500 0.09016 0.1099 2.200 0.4245 1000 0.06843 0.1126 1.973 0.4138 1500 0.05558 0.1142 1.851 0.4074 2000 0.04639 0.1152 1.768 0.4028 2500 0.03924 0.1160 1.706 0.3992 3000 0.03336 0.1166 1.657 0.3962 3500 0.02838 0.1172 1.617 0.3937 4000 0.02406 0.1176 1.583 0.3914 4500 0.02023 0.1180 1.553 0.3895 5000 0.01680 0.1184 1.527 0.3877 10 91.2 0.0756 0.1040 2.710 0.4561 250 0.0446 0.1081 2.305 0.4397 500 0.0230 0.1108 2.062 0.4280 750 0.0102 0.1122 1.931 0.4211 1000 0.0011 0.1132 1.843 0.4160 1250 -0.0060 0.1140 1.778 0.4121 1500 -0.0118 0.1146 1.726 0.4089 1750 -0.0167 0.1151 1.683 0.4061 2000 -0.0210 0.1155 1.646 0.4037 2500 -0.0281 0.1162 1.587 0.3997 3000 -0.0341 0.1167 1.540 0.3964 20 250 0.02850 0.1084 2.269 0.4406 500 0.00685 0.1109 2.029 0.4286 750 -0.00592 0.1123 1.899 0.4214 1000 -0.01504 0.1133 1.812 0.4162 1250 -0.02213 0.1140 1.747 0.4122 1500 -0.02795 0.1146 1.695 0.4089 1750 -0.03288 0.1150 1.653 0.4061 2000 -0.03716 0.1154 1.617 0.4036 2500 -0.04433 0.1161 1.558 0.3995 3000 -0.05020 0.1166 1.511 0.3961

Table 1: The scale dependence of C1(Q)- C4(Q) for minimal supergravity when mt = 175 GeV for tanβ = 2, 5, 10 and 20.

17 Minimal Supergravity µ< 0 ± ± 0 tan β Φ H u˜l e˜l e˜r t˜1 t˜2 g˜ h χ˜1 χ˜2 χ˜1 m0 2 5 326 290 212 207 264 325 316 69 102 224 48 204 10 479 419 320 315 353 429 459 78 139 303 67 315 20 687 598 463 459 483 579 649 86 190 419 94 459 5 2.5 318 352 292 285 265 352 295 97 77 180 42 282 5 594 589 560 556 365 507 425 103 114 232 60 556 10 930 906 888 884 510 744 610 109 167 309 86 886 20 1417 1381 1368 1365 742 1113 873 116 243 423 123 1368 10 2.5 416 464 403 393 323 431 316 106 73 184 42 395 5 3702 4089 3914 3887 2311 3311 1272 136 190 382 158 3920 10 5963 6714 6365 6318 3855 5428 2776 144 283 797 273 6370 20 8875 10536 9622 9527 6170 8616 4945 150 404 1409 400 9596 20 2.5 1889 2136 2044 2003 1202 1697 566 128 104 214 69 2080 5 3581 4198 3906 3827 2480 3383 1764 138 194 515 178 3980 10 5540 6585 6114 5978 3893 5270 3124 145 282 895 274 6210 20 8007 10092 8954 8734 6078 8167 5322 151 403 1516 399 9060

Table 2: The upper bound on sparticle masses for minimal supergravity when mt = 175 GeV and µ < 0 for diﬀerent values of tanβ and ﬁne tuning measure Φ0. All the masses are in GeV.

18 ′ tan β δ1 δ2 C1 2 -1.0 1.0 -0.341 -0.75 0.75 -0.067 -0.5 0.5 0.208 -0.25 0.25 0.483 0.0 0.0 0.757 0.25 -0.25 1.032 0.5 -0.5 1.306 0.75 -0.75 1.581 1.0 -1.0 1.855 5 -1.0 1.0 -0.572 -0.75 0.75 -0.393 -0.5 0.5 -0.215 -0.25 0.25 -0.036 0.0 0.0 0.142 0.25 -0.25 0.321 0.5 -0.50 0.499 0.75 -0.75 0.677 1.0 -1.0 0.856 10 -1.0 1.0 -0.597 -0.75 0.75 -0.429 -0.5 0.5 -0.261 -0.25 0.25 -0.092 0.0 0.0 0.076 0.25 -0.25 0.244 0.5 -0.5 0.412 0.75 -0.75 0.580 1.0 -1.0 0.748 20 -1.0 1.0 -0.603 -0.75 0.75 -0.437 -0.5 0.5 -0.272 -0.25 0.25 -0.106 0.0 0.0 0.060 0.25 -0.25 0.225 0.5 -0.5 0.391 0.75 -0.75 0.556 1.0 -1.0 0.722

′ Table 3: C1(MZ ) for diﬀerent values of δ1 and δ2 when mt = 175 GeV for tanβ = 2, 5, 10 and 20.

19 Non-universal case: (δ1, δ2, δ3, δ4) = (1, 1, 0, 0), µ< 0 − ± ± 0 tan β Φ H u˜l e˜l e˜r t˜1 t˜2 g˜ h χ˜1 χ˜2 χ˜1 m0 2 5 313 291 148 140 267 328 319 70 104 225 49 134 10 457 419 220 207 354 430 459 79 140 304 68 205 20 655 601 317 296 488 585 656 87 193 420 95 299 5 2.5 213 274 133 121 243 336 301 95 79 181 44 109 5 356 391 221 228 324 430 429 103 115 233 62 204 10 535 569 334 312 462 580 620 110 170 310 89 315 20 775 841 485 451 677 820 915 116 254 425 130 462 10 2.5 203 285 129 103 246 349 316 104 74 185 43 93 5 371 406 234 216 333 447 446 110 112 237 62 215 10 559 583 357 332 471 598 637 116 169 314 90 338 20 810 843 520 483 680 828 920 122 251 427 130 496 20 2.5 216 286 125 69 242 350 315 105 69 186 40 75 5 383 409 236 203 330 451 448 111 109 237 60 216 10 577 590 357 323 472 604 646 117 167 315 89 343 20 843 854 519 479 687 835 932 122 252 428 130 508

Table 4: The upper bounds on sparticle masses for the case of non-universalities in the Higgs sector when (δ1, δ2) = (1, 1), δ3 =0= δ4, mt = 175GeV , and µ < 0 for diﬀerent values of tanβ and Φ. All the masses− are in GeV.

20 (δ1, δ2, δ3, δ4) = ( 1, 1, 0, 0) ′ − tan β Q(GeV ) C1 C2 C3 C4 2 91.2 -0.341 0.071 4.284 0.312 250 -0.363 0.0765 3.742 0.3110 500 -0.378 0.0802 3.415 0.3101 750 -0.387 0.0825 3.239 0.3094 1000 -0.394 0.0841 3.119 0.3089 1250 -0.399 0.0853 3.030 0.3084 1500 -0.404 0.0863 2.959 0.3080 1750 -0.408 0.0872 2.901 0.3077 2000 -0.411 0.0879 2.851 0.3073 5 91.2 -0.572 0.1024 2.871 0.4491 250 -0.602 0.1069 2.452 0.4348 500 -0.624 0.1099 2.200 0.4245 750 -0.636 0.1115 2.064 0.4183 1000 -0.645 0.1126 1.973 0.4138 1250 -0.652 0.1135 1.905 0.4103 1500 -0.658 0.1142 1.851 0.4074 1750 -0.663 0.1147 1.806 0.4049 2000 -0.667 0.1152 1.768 0.4028 10 91.2 -0.597 0.1040 2.710 0.4561 250 -0.628 0.1081 2.305 0.4397 500 -0.649 0.1108 2.062 0.4280 750 -0.662 0.1122 1.931 0.4211 1000 -0.671 0.1132 1.843 0.4160 1250 -0.678 0.1140 1.778 0.4121 1500 -0.684 0.1146 1.726 0.4089 1750 -0.689 0.1151 1.683 0.4061 2000 -0.693 0.1155 1.646 0.4037 20 91.2 -0.603 0.1043 2.671 0.4575 250 -0.634 0.1084 2.269 0.4406 500 -0.655 0.1109 2.029 0.4286 750 -0.668 0.1123 1.899 0.4214 1000 -0.677 0.1133 1.812 0.4162 1250 -0.684 0.1140 1.747 0.4122 1500 -0.690 0.1146 1.695 0.4089 1750 -0.695 0.1150 1.653 0.4061 2000 -0.699 0.1154 1.617 0.4036

′ Table 5: The scale dependence of C1(Q),C2(Q) – C4(Q) for (δ1, δ2, δ3, δ4)=( 1, 1, 0, 0) for − mt = 175 GeV and tanβ = 2, 5, 10 and 20.

21 (δ1, δ2, δ3, δ4) = (0, 0, 1, 0), µ< 0 ± ± 0 tan β Φ H u˜l e˜l e˜r t˜1 t˜2 g˜ h χ˜1 χ˜2 χ˜1 m0 2 5 290 292 162 140 266 326 319 70 104 225 49 147 10 418 419 242 211 351 429 456 79 139 304 68 226 20 597 601 349 306 486 583 656 87 193 420 95 331 5 2.5 198 275 151 125 244 337 301 95 79 181 44 126 5 325 391 258 223 325 430 429 103 115 233 62 239 10 485 559 389 340 453 573 611 110 168 310 87 368 20 702 807 571 501 652 791 880 116 245 424 125 546 10 2.5 194 291 144 110 247 349 316 104 74 185 43 110 5 342 422 279 239 333 445 446 110 112 237 62 259 10 514 606 428 371 471 599 637 116 169 314 90 407 20 747 870 629 549 680 828 920 122 251 427 130 604 20 2.5 213 293 134 77 245 352 316 106 70 187 41 88 5 360 431 281 229 331 452 449 112 110 238 61 261 10 541 621 433 364 473 605 646 118 168 316 90 416 20 804 894 638 543 688 836 932 123 252 429 131 620

Table 6: The upper bound on sparticle masses for non-universalities in the third generation when (δ3, δ4) =(1, 0), mt = 175GeV , and µ< 0 for diﬀerent values of tanβ and Φ. All the masses are in GeV.

22 √s = 183 GeV LEP 95% C.L.lower bound on mh tan β mass lower bdd (GeV) Φ(µ< 0) 2 86 (L3) 20 74 (OPAL scan B) 8 88 (ALEPH) 23 84 (DELPHI) 18 5 72 (L3) < 2 71 (OPAL scan B) 73 (ALEPH) 76 (DELPHI) 10 72 (L3) < 2 70 (OPAL scan B) 76 (ALEPH) 75 (DELPHI) 20 71 (L3) < 2 70 (OPAL scan B) 76 (ALEPH) 76 (DELPHI) √s = 183 GeV LEP 95% C.L. lower bounds on various sprticles masses Particle mass lower bdd (GeV) Φ(µ< 0) 24 independent of m0 (DELPHI) 0 χ 14 any m0 (ALEPH) < 1.5 for tan β 2 ≥ 27 for tan β = 2 (L3) < 1.5 χ± 51 (ALEPH) < 1.5 for tan β 2 ≥ t˜ t˜ cχ m˜ > 74 (ALEPH) < 1.5 for tan β 2 → t ≥ t˜ blνχ m˜ > 82 (ALEPH) → t 95% C.L. lower bounds on various sprticles masses from ref[34] Particle mass lower bdd (GeV) Φ(µ< 0) ± χ mχ± > 45, .66 pb < 1.5

mχ± > 124, .01 pb Φ > 8, tan β = 2 Φ > 5.8, tan β = 5 q˜ g˜ mg˜ > 230, heavy squarks Φ > 2.7, tan β = 2

mq,˜ g˜ > 260, mq˜ = mg˜ Φ > 4.0, tan β = 2 Φ < 1.8, tan β 5 ≥

mq˜ > 219, heavy gluinos Φ > 2.8, tan β = 2 Φ < 1.8, tan β 5 ≥ Table 7: Current experimental lower bounds on masses of the lightest Higgs and various sparticles from LEP and the Tevatron. Corresponding ﬁne-tunings (µ< 0) are also shown.

23 Fig. 1a Fig. 1c β=2 µ<0 β=2 µ<0 mt=175 GeV tan mt=175 GeV tan 500 600

Φ ~ 0 > 20 t 1 500 ~ 400 t 2 e~

400 Φ 300 0 > 10

300 ~

[GeV] LHC energy reach on l L 0 Φ m [GeV]

m 200 0 > 5 LHC energy reach on l~ 200 R Φ ~ 0 > 2.5 TeV33 energy reach on t 1 100 100

0 0 50 100 150 200 250 0 5 10 15 20 Φ m1/2 [GeV] 0 Fig.1a. Contour plot of the upper limit in Fig.1c. Upper bounds on mass of thee ˜L, the m0 m 1 plane for diﬀerent values of Φ0 of the light stop t˜1, and of the heavy stop − 2 ˜ when mt = 175 GeV, tan β = 2 and µ < t2 for the same parameters as in Fig.1a. 0. The allowed region lies below the curves. Fig. 1d β=2 µ<0 mt=175 GeV tan Fig. 1b 1000 0 m =175 GeV tanβ=2 µ<0 h t ∼± 1500 χ 2 χ∼± ∼01 ~ ~ ≈ ~ LHC energy reach on m (2m m ) χ 1 g g q H0 ~ −1 u L MI 2fb limit on chargino g~

1000 0 100 TeV33 energy reach on h

m [GeV] TeV 100pb−1 limit on chargino

−1 m [GeV] TeV 20pb limit on chargino −1 ~ MI 25fb energy reach on mg 500

10 0 5 10 15 20 Φ 0 0 0 5 10 15 20

Φ0 Fig.1d. Upper bounds on masses of the light 0 ± Fig.1b. Upper bounds on mass of the heavy Higgs h , of the light charginoχ ˜1 , of the 0 ± 0 Higgs H , of the gluino and of the squarku ˜L heavy charginoχ ˜2 , and of the neutralinoχ ˜1 (for the ﬁrst two generations) for the same for the same parameters as in Fig.1a. parameters as in Fig.1a.

24 Fig. 2a Fig. 2c β=5 µ<0 β=5 µ<0 mt=175 GeV tan mt=175 GeV tan

1400 ~ t 1 Φ 1200 ~ t 2 0 >20 1200 e~

1000

Φ 800 800 0 >10 [GeV] 0 m [GeV] m 600 Φ 0 >5 400 400 ~ Φ LHC energy reach on l L 0 >2.5 LHC energy reach on l~ 200 R ~ TeV33 energy reach on t 1

0 0 50 100 150 200 250 300 350 0 5 10 15 20 Φ m1/2 [GeV] 0 Fig.2a. Contour plot of the upper limit in Fig.2c. Upper bounds on mass of thee ˜L, the m0 m 1 plane for diﬀerent values of Φ0 of the light stop t˜1, and of the heavy stop − 2 ˜ when mt = 175 GeV, tan β = 5 and µ < t2 for the same parameters as in Fig.2a. 0. The allowed region lies below the curves. Fig. 2d β=5 µ<0 mt=175 GeV tan Fig. 2b 1000 β=5 µ<0 0 mt=175 GeV tan h χ∼± 2000 2 χ∼± ~ ~ ≈ ~ ∼01 LHC energy reach on mg (mg mq ) χ 1 MI 2fb−1 limit on chargino

~ ~ ≈ ~ 1500 LHC energy reach on mg (2mg mq ) TeV33 energy reach on h0 0 H 100 ~ u L ~ 1000 g m [GeV] TeV 100pb−1 limit on chargino

m [GeV] TeV 20pb−1 limit on chargino

500

−1 ~ MI 2fb energy reach on mg 10 0 5 10 15 20 Φ 0 0 0 5 10 15 20

Φ0 Fig.2d. Upper bounds on masses of the light 0 ± Fig.2b. Upper bounds on mass of the heavy Higgs h , of the light charginoχ ˜1 , of the 0 ± 0 Higgs H , of the gluino and of the squarku ˜L heavy charginoχ ˜2 , and of the neutralinoχ ˜1 (for the ﬁrst two generations) for the same for the same parameters as in Fig.2a. parameters as in Fig.2a.

25 Fig.3a Variation with scale Q0 Fig.3b Variation with scale Q0 β µ<0 β µ<0 tan =10 A0=0 m0=2 TeV m1/2=200 GeV tan =20 A0=0 m0=2 TeV m1/2=200 GeV 400 400 µ µ tree tree µ µ tot 200 tot 200 ∆µ ∆µ

0 0 [GeV] [GeV] −200 −200

−400 −400

−600 −600 500 1000 1500 2000 400 600 800 1000 1200

Q0 [GeV] Q0 [GeV] Fig.3a. Variation of µ with the scale Q0 where Fig.3b. Variation of µ with the scale Q0 where the minimization of the potential is carried the minimization of the potential is carried out for the case when tan β = 10, A0 = 0, out for the case when tan β = 20, with the m0 = 2000GeV, m1/2 =200 GeV and µ< 0. other parameters the same as in Fig.3a.

26 2 α2 2 β2 2 α2 2 β2 fig.4a m’1/2 / + m0 / = 1 fig.4c m’1/2 / − m0 / = 1 Φ=10; A0=0; (C1 C2 C3 C4) = (0.757 0.0711 4.28 0.312) (C1 C2 C3 C4) = (−0.034 0.117 1.54 0.396) 500 3000 A = −500 A0= 500 A = 0 0 0 α 400 Φ α= 215 α= 235 = 215 =20 β β= 1446 β= 1582 = 1446 α=198 2000 300 β=472

200 Φ=2.5 α=73 1000 100 β=174

0 0 [GeV] [GeV] 0 0

m −100 m −1000 −200

−300 −2000 −400

−500 −3000 −300 −200 −100 0 100 200 300 100 200 300 400 500

m1/2 [GeV] m1/2 [GeV] Fig.4a. Diagrammatic illustration of the Fig.4c. Diagrammatic illustration of the hy- ellipse represented by Eq.(12), where the perbola represented by Eq.(14) and Eq.(28), values of C1 – C4 are for tanβ=2 and where the values of C1 – C4 are for tanβ=10 Q = MZ from Table 1. The rele- and Q=3000 GeV from Table 1. The relevant parts of the ellipses are in solid line. vant parts of the hyperbolae are in solid line.

2 α2 2 β2 2 α2 2 β2 fig.4b m’1/2 / − m0 / = 1 fig.4d −m’1/2 / + m0 / = 1

A0=0; (C1 C2 C3 C4) = (−0.034 0.117 1.54 0.396) (C1 C2 C3 C4) = (−0.034 0.117 1.54 0.396) 6000 7000

A0=−3000 A0= 3000 GeV 6000 Φ Φ= 2.5 4000 = 2.5

5000 2000 Φ=2.5 10 20 4000 0 [GeV] [GeV] 0 0 A0=−3000 3000 A0= 3000 m m Φ Φ= 20 = 20 −2000 2000

−4000 1000

−6000 0 0 200 400 600 800 1000 −1400 −1000 −600 −200 200 600 1000 1400

m1/2 [GeV] m1/2 [GeV] Fig.4b. Diagrammatic illustration of the hy- Fig.4d. Diagrammatic illustration of the hyperbola represented by Eq.(14) and Eq.(28), perbola represented by Eq.(19) and Eq.(30), where the values of C1 – C4 are for tanβ=10 where the values of C1 – C4 are for tanβ=10 and Q=3000 GeV from Table 1. The relevant and Q=3000 GeV from Table 1. The relevant parts of the hyperbolae are in solid line. parts of the hyperbolae are in solid line.

27 fig.5a Scale dependence of C1

mSUGRA mt=175 GeV 0.8

tanβ 0.7 = 2

0.6

0.5

0.4 (Q) 1

C 0.3

0.2

0.1 5

10 0.0 20

−0.1 100 1000 10000 Q [GeV] Fig.5a. The scale dependence of C1(Q) for minimal supergravity when mt = 175 GeV for tanβ = 2, 5, 10 and 20.

28 fig.5b tanβ=10 Φ=10 µ<0 8000

All A A =−2000 GeV A =−1000 GeV 6000 0 0 0

4000 [GeV] 0 m 2000

8000

A = 0 A = 500 GeV A = 1000 GeV 6000 0 0 0

4000 [GeV] 0 m 2000

0 200 400 600 800 200 400 600 800 200 400 600 800 1000

m1/2 [GeV]

Fig.5b. Allowed region in the m0 – m1/2 plane in the minimal supergravity case for mt = 175 GeV, tanβ=10, Φ0 =10 and negative µ.

29 fig.5c tanβ=20 Φ=10 µ<0 8000

6000

4000 [GeV] 0 m A =−500 GeV All A A0=−2000 GeV 0 2000 0

8000

6000

4000 [GeV] 0 m 2000 A = 0 A = 500 GeV 0 0 A0=1500 GeV

0 200 400 600 800 1000 200 400 600 800 1000 200 400 600 800 1000

m1/2 [GeV]

Fig.5c. Allowed region in the m0 – m1/2 plane in the minimal supergravity case for mt = 175 GeV, tanβ=20, Φ0 =10 and negative µ.

30 fig.6 (δ1,δ2)=(−1,1) tanβ=2 Φ=10 µ<0 3000

2500 All A A = −1500 GeV A = −500 GeV 2000 0 0 0

1500 [GeV] 0

m 1000

500

3000

2500 A = 0 A = 500 GeV A = 1500 GeV 2000 0 0 0

1500 [GeV] 0

m 1000

500

0 0 200 400 600 200 400 600 200 400 600

m1/2 [GeV]

Fig.6. Allowed region in the m0 – m1/2 plane under the non-universal boundary condition of (δ1, δ2)=(-1,1) for mt = 175 GeV, tanβ=2, Φ =10 and negative µ.

31 Fig. 7 µ<0 mt=175 GeV 130

TeV33 25fb−1 limit on m 120 h

110 > 20 Φ 0 > 10 Φ 0 100 > 5 Φ 0

[GeV] > 2.5 h Φ 0

m 90

60 1 2 3 4 5 6 7 tanβ 0 Fig.7. Upper bounds on the light Higgs h mass for diﬀerent values of Φ0 as a function of tan β when mt = 175 GeV and µ< 0.